Some Aspects of Growth-Fragmentation

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Some Aspects of Growth-Fragmentation

Dissertation zur

Erlangung der naturwissenschaftlichen Doktorwürde (Dr. sc. nat.)

vorgelegt der

Mathematisch-naturwissenschaftlichen Fakultät der

Universität Zürich von

Benjamin Dadoun aus Frankreich

Promotionskommission Prof. Dr. Jean Bertoin (Vorsitz) Prof. Dr. Bénédicte Haas Prof. Dr. Bastien Mallein Prof. Dr. Rémi Abgrall Prof. Dr. Ashkan Nikeghbali

Zürich, 2019

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Some Aspects of Growth-Fragmentation

Benjamin Dadoun Ph. D. thesis

Abstract

This thesis treats stochastic aspects of fragmentation processes when growth and/or immigration of particles are incorporated as a compensating phenomenon. In a first part, we study the asymptotic behavior of self-similar growth-fragmentation processes, extending the results related to pure fragmentations. In a second part, we prove that self-similar growth-fragmentations arise as scaling limits of truncated Markov branching processes and we provide a rather general criterion. This bolsters the conviction that growth-fragmentations appear in many discrete Markovian structures, as already observed in random planar geometry. Lastly, we study a growth-fragmentation with immigration equation. In particular, we investigate the asymptotic behavior of the solution by relating it to a stochastic particle system in which immigrate copies of a certain growth-fragmentation process.

i

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À mes frères,

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iii

Chaque fois que la science avance d’un pas, c’est qu’un imbécile la pousse,

sans le faire exprès.

Émile Zola,La Joie de vivre

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Acknowledgments vii

General introduction 1

0.1 Fragmentation processes . . . 4

0.1.1 General framework . . . 4

0.1.2 Pure fragmentations . . . 5

0.1.3 Growth-fragmentation processes . . . 7

0.2 Asymptotics of self-similar fragmentations . . . 15

0.2.1 Homogeneous fragmentations . . . 15

0.2.2 Self-similar fragmentations . . . 17

0.3 Self-similar fragmentations as scaling limits. . . 20

0.4 Fragmentation equations . . . 24

1 Asymptotics of self-similar growth-fragmentation processes 29 1.1 Introduction . . . 29

1.2 Compensated fragmentations . . . 32

1.2.1 Prerequisites. . . 32

1.2.2 Uniform convergence of the additive martingales . . . 37

1.2.3 On the largest fragment . . . 44

1.2.4 On abnormally large fragments . . . 46

1.3 Self-similar growth-fragmentations . . . 48

1.3.1 Prerequisites. . . 48

1.3.2 Convergence of the empirical measure . . . 53

1.3.3 Asymptotic behavior of the largest fragment . . . 57

1.3.4 Freezing the fragmentation . . . 60

2 Self-similar growth-fragmentations as scaling limits of Markov branching processes 65 2.1 Introduction . . . 65

2.2 Assumptions and results . . . 68

2.3 Convergence of finite-dimensional marginals . . . 73

2.4 A size-biased particle and a many-to-one formula . . . 75

2.5 Proof of Theorem 2.2.1 . . . 79

2.6 Proof of Theorem 2.2.2 . . . 81 3 A growth-fragmentation with immigration equation 85

iv

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CONTENTS v

3.1 Introduction . . . 85

3.2 Existence . . . 87

3.3 Uniqueness . . . 88

3.4 Comparison of solutions . . . 89

3.5 The growth-fragmentation-immigration process . . . 91

3.6 Population-dependent immigration . . . 93

Bibliography 97

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Acknowledgments Remerciements

À vous, Jean, en premier lieu. Merci de votre disponibilité et de votre patience infinie nonobstant mes trop nombreux moments de doute et passages à vide. Merci également à Bastien et à Bénédicte, d’avoir scrupuleusement rapporté toutes les bétises que j’ai pu écrire

— à Rémi et à Ashkan de faire aussi partie du jury.

Second obligatory thanks go to Switzerland and to the city of Zurich especially. How lucky I was to live in this fantastic place!

.

Silvia, my dear officemate and academic twin sister, thank you for the reliable companionship.

Like real twins we developed our own secret language to share so many crucial things, and I leave here a message that only you (I hope!) should be able to decipher: t.m.l.s.d.p.a.u, e.d.t.r. !

(Zut, ça ne me semble pas aussi indéchiffrable que ça...)

Not forgetting here Marcel, whom we forgot for lunch too many times... I admire your enthusiasm and curiosity. Thanks for all the nice seminars, debates and (often weird) discussions, for the ice cream breaks and the best fondue and movie evenings at your place.

I would also like to thank my other academic siblings and close colleagues, Cyril, Quan, Gabriel, François, Benedetta, Robin, Herry, Emma, Martina, Gabriele, Jungtaek... as well as the people from the probability group at ETH, Mayra, Yilin, Angelo, Avelio, Titus, Juhan, Alberto, Maximilian, Daniel... Kind thoughts also toward other officemates, colleagues, and professors from other groups at the institute, Philipp, Élise, Davide, Paula, Barbara, Maria, Jianfang, Helen, Jacopo, Jehanne, Raúl, Karan, Andres, Genta, Francesco, David, Valentin, Mathilde... (The previous lists are not exhaustive.) Thanks for all these sweet and relaxing breaks and for the “drinking seminars” and dinners at restaurants and bars of all sorts.

To my badminton, tennis, and ping-pong friends, Timo, Henri, Anne, Lucas, Esther, Gilbert, Chang, Nicola, Macarena, Michael, Sebastian, Yifan, and many others, including the coaches.

Merci à Charlène de m’avoir initié au ski dans les Alpes suisses, et à Aser de savoir encore plus mal skier que moi¹. Warm thanks to Céline who introduced me to snowbarding. And my deep gratitude to everyone who helped me and cheered me up when I stupidly fragmented my shoulder at the beginning of the summer 2018 — to my physiotherapist Oliver who led

1J’ai une preuve vidéo incontestable.

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me to the best possible recovery.

Also, it was a pleasure to be surrounded by so friendly and devoted secretaries, IT gurus and cleaning personal. To Franziska, Grit, Bettina, Claudia, Maria, Carsten, Benjamin, and all the trainees: thank you.

Naturellement, j’en viens à remercier mes amis de plus longue date, Pierre, Ludo, Alain, Lia, Romain, Claıre, Jessica, Laurent, Édouard, Jill-Jênn, Tristan, Lilian, mais aussi Alexandre,^♥ Romain, Arthur... J’embrasse enfin mes proches, papa, maman, Élie, Michaël, Jonathan, Liliane, Audrey, Julie, et bien sûr Juna. Je vous dédie cette thèse qui grâce à votre soutien inconditionnel a été un agréable moment de ma vie. Comme chantait Alain Souchon :

Si tout est moyen

Si la vie est un film de rien Ce passage-là était vraiment bien Ce passage-là était bien.

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Introduction

The word fragmentation designates a process in which an object is shattered into many smaller pieces called fragments. One may think of those resulting from the collision between two celestial objects of comparable sizes, or from the fragmentation of a meteorite entering the atmosphere. In cell biology, one may picture mitosis as a kind of fragmentation. Other instances where fragmentations occur include earthquakes in geophysics, fission in biology or in nuclear physics, sequencing in mass spectrometry, crushing in mineral processing, data transmission in telecommunications, etc.

As a simple and first, chronological example, let us take a pile of (fine) sand, split it in a truly random way, and repeat the operation independently on each two subpiles. Obviously, there are 2ⁿ piles after the n-th iteration. But what is the repartition of their respective volumes? Such a question was answered in 1941 by A. N. Kolmogorov, in the paper [74]

“Übers logaritmisch normale Verteiligsgsetz vo de Dimensione vo de Teili bi de Zerszücklig”.

As the title may suggest, the logarithms of the volumes follow a Normal distribution (the well-known bell-shaped curve) when n is sufficiently large; see Figure 0.1a for a simulation.

More generally, Kolmogorov’s result applies to the situation where “particles” may be divided into a random number of fragments with random “sizes”, possibly in a non-conservative, but always homogeneous, way.

The question of inhomogeneous fragmentation, which he asked at the end of his work, addresses the situation where the rate of grinding may depend on the particle size (referring to our example, the splitting scheme can then be a function of the volume of sand being split). More precisely, one could imagine that particles with size x fragmentate x^α “faster”

than particles with size 1 (so the original formulation is enclosed in the homogeneous case

-30 -25 -20 -15 -10

0.00 0.02 0.04 0.06

0.08 Logarithmic volume

repartition of sandpiles The Normal(−n, n) density distribution

Figure 0.1a. Empirical distribution of sandpiles aftern = 20 steps.

1

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0.00002 0.00004 0.00006 0.00008 0.00010 0

20 000 40 000 60 000 80 000

Volume repartition of sandpiles The generalized-Gamma(t⁻¹^/^α,1, α) density distribution

Figure 0.1b. Empirical distribution of sandpiles at large time for α=¹/2.

α = 0). Kolmogorov predicted that in the nowadays called self-similar case α 6= 0, “the logarithmic normal law is no longer applicable”. This discrepancy was confirmed twenty years later by one of his former students, A. F. Filippov² [60]. If u_t(x) denotes the average density of sandpiles of volume xat time t, then u_t solves the integro-differential equation

∂

∂tu_t(x) =−x^αu_t(x) + 2 Z ∞

x

y^α−1u_t(y) dy. (0.1)

For α >0, the analysis of this equation shows that whent is sufficiently large,u_t is close to a generalized Gamma distribution³; see Figure 0.1b. For α < 0, Filippov shone a light on the phenomenon of “formation of dust”: because particles fragmentate at higher rates as they get smaller, the total mass of the system decreases continuously and vanishes within a finite amount of time (although the mass involved in each individual fragmentation is conserved).

There is a natural genealogy induced by a fragmentation process, where “daughter particles” are related to the “mother particle” they originate from. In the self-similar setting, we easily picture a sort of phylogenetic tree where the length of a mother-daughter branch is proportional to the daughter’s lifetime. When α < 0, because of the total extinction of mass at large times, this fragmentation tree is “compact”. In many cases (typically, when the “macroscopic” fragmentations into large comparable fragments are rare), its global shape does further not depend heavily on the exact splitting scheme, and we often observe so called continuum random trees, which bear interesting fractal properties. See Figure 0.2.

Fragmentation equations like (0.1) appear in a large variety of biological or physical models of particle systems and constitute an object of study as a whole, generally (at least, originally) from a non-probabilistic point of view. Besides the customary questions of existence and uniqueness of solutions, one typically wants to describe the stationary regime

2Incidentally, Filippov was twenty years younger than Kolmogorov, whom he acknowledges for his assistance and advice. Both had the same academic grandfather, D. F. Egorov.

3For this simple example, Filippov even provides a semi-explicit expression foru_tat any timet.

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3

Figure 0.2. Large fragmentation tree with binary splitsn 7→ {k, n−k}, 1≤k < n,occurring at rate ∝n^α for α= ⁻¹/2, and with probability∝ ⁿ_k R1

1/2x^k−³^/²(1−x)^n−k−³^/²dx.

and estimate the speed of convergence toward the asymptotic profile. Equilibrium may arise when fragmentation is compensated by growth of particles or immigration of new particles.

This thesis treats some of the aforementioned aspects when growth and possibly immigration is added to the picture. In a growth-fragmentation process, particles are not only subject to (random) fragmentation but may as well grow larger or smaller in a continuous (also random) way. In Chapter 1, we look at asymptotics of empirical measures associated with the fragments, extending prior results in this vein (Kolmogorov [74], Filippov [60], Baryshnikov and Gnedin [16], Bertoin and al. [23, 26, 30, 32], Kyprianou et al. [77]). Then, in Chapter 2, motivated by the appearance of self-similar growth-fragmentation processes in random planar geometry [25, 24, 81], we consider a simple Markovian model acting as discrete growth-fragmentation and we discuss its self-similar scaling limit by amending a criterion of Bertoin and Kortchemski [27]. This work can be seen as a humble addition to the general and various criteria of Haas, Miermont et al. [86, 65, 87, 66, 63] for self-similar pure-fragmentations. Lastly, in Chapter 3, we focus on a growth-fragmentation equation as studied by Bertoin and Watson [34, 20, 36] but with an additional term accounting for immigration.

In the remaining of this general introduction, we formally set up the growth-fragmentation processes and present more precisely the different questions that we alluded above.

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0.1 Fragmentation processes

A first, fundamental property of fragmentation processes is the so calledbranching property: fragments split and evolve independently of one another. As such, we may define them within the framework of branching processes on (0,∞), which we now introduce.

0.1.1 General framework

For any measurable function f: (0,∞) → (0,∞), we write f ∈ B_0< if it is bounded from below away from zero, that is inf{f(x) : x ≥ a} > 0 for every a > 0. We set f(0) := 0 and let M_f be the space of non-increasing sequences x:= (x1, x₂, . . .) on[0,∞) such that

hx, fi:=X

j≥1

f(x_j)<∞.

(Null values inxare disregarded; their presence is merely needed to include finite sequences.) We see M_f as a subspace of M₀, the space of non-increasing null sequences endowed with the topology of pointwise convergence, which because of monotonicity is also that of uniform convergence. Equivalently, we may view elements x ∈ M₀ as point measures P

j≥1δ_x_j on(0,∞)under the topology of vague convergence. This makesM₀ a Polish space which we naturally equip with its Borel σ-field.

For the remaining of this section, we fixf ∈ B0<∪ {0}and note that for point measuresx and x⁽ⁱ⁾, i ≥ 1, we can always define the scalar multiplication λx := P

j≥1δ_λx_j, λ > 0, and the sum U

i≥1x⁽ⁱ⁾ :=P

i,j≥1δ_x(i) j

(not necessarily in M₀).

Definition 0.1.1. Let X := (X(t) : t ≥ 0) be a M_f-valued stochastic process, which is continuous in probability and whose conditional law given X(0) =δ_x, x > 0,is denoted P_x.

• We call Xabranching process (inM_f) if it has the (temporal)branching property: for every s ≥ 0, the family (X(t+s) : t ≥ 0) given X(s) = (x1, x2, . . .) is independent of (X(r) : r ≤s)and distributed like(U

i≥1X⁽ⁱ⁾(t) : t≥0), where theX⁽ⁱ⁾are independent processes with law P_x_i, respectively.

• If further there exists α ∈ R such that for every x > 0, the law of (xX(x^αt) : t ≥ 0) under P₁ is P_x, then we say that X is self-similar; α is the index of self-similarity.

When α = 0, the process is said homogeneous.

Additive martingales and supermartingales associated with branching processes constitute one essential tool. Let us state a first elementary fact in this direction. Hereafter,E_x denotes the expectation under P_x.

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0.1. FRAGMENTATION PROCESSES 5 Proposition 0.1.2 (Corollary 1 in [19]; see also Biggins and Kyprianou [42]). Let C > 0 and suppose that f is C-excessive for X, in the sense that for all x >0, t≥0,

E_x

hX(t), fi

≤C^tf(x)

(when there is equality, we rather say that f is C-invariant). Then the process C^−thX(t), fi, t≥0,

is a supermartingale under P_x for everyx >0. (It is in fact a martingale iff isC-invariant.) Often, typically for self-similar fragmentations, we restrictB_0<to power functionsf_q: x7→x^q with q ≥0and identify M_f_q as a closed subspace of the sequence space `^q(N).

0.1.2 Pure fragmentations

Definition 0.1.1embraces processes that may be far from the intuitive notion of fragmentation.

In a truly (i.e., pure) fragmentation process, one indeed also expects X(t) to be “finer”

than X(s)for all t ≥s.

Definition 0.1.3. A branching process is called a pure-fragmentation process (or simply a pure fragmentation), if its sample paths are non-increasing with respect to the lexicographic ordering ≤_lex onM₀.

It is clear from the branching property that a branching process is a pure fragmentation if and only if under P_x, x > 0,it has values in the subspace

Sx :=n

s∈ M₀: s≤_lex(x,0, . . .)o .

We call s∈S :=S1 aconfiguration, giving the different arrangementsx·s:= (xs1, xs2, . . .) of fragments which may result from the fragmentation of a particle with initial size x > 0.

This includes the trivial configuration 1 := (1,0, . . .) and the dust 0 := (0,0, . . .). Often, the size parameter x rather corresponds to a “mass” or any quantity which cannot increase under fragmentation. Then the total masshX(t), f1iis also non-increasing over time, so that under P_x, x >0, the sample paths have values in {x·p: p∈P}, where

P :=

(

p∈ M₀: X

i≥1

p_i ≤1 )

is the space ofmass-partitions. In this situation, f₁ is1-excessive forX(soP

i≥1X_i(t), t≥0, is a supermartingale). Of course, there are other situations where this is not true, for instance when the size of a particle is measured by its diameter.

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Note that a fragmentation eventx7→x·pspecified byp∈P may induce a loss of mass, that is a possibly positive fraction p0 := 1−P

i≥1pi of the mass x may be reduced to dust.

We say that p (resp. a measure ν on P) is conservative if p₀ = 0 (resp. ν(p₀ >0) = 0).

In the next sections we briefly sketch two constructions of pure fragmentations, and forward the interested reader to the first three chapters of the monograph [21].

0.1.2.1 Fragmentation chains

Let ν_x, x >0, be a finite measure on M₀, with support in Sx. We assume that the family (ν_x)_x>0 depends in a measurable way on the variable x. A (pure-)fragmentation chain with kernel (ν_x)_x>0 is a system of non-interacting particles which, at rate ν_x(Sx) according to their respective size x, are each replaced by a cloud of particles with law ν_x(·)/ν_x(Sx).

It is formally defined as a continuous-time branching Markov chain (X(t) : t ≥ 0) with intensity kernel q onM₀ given by q(0,·) := 0 and

q(x,dy) :=X

i≥1

νxi(y+δxi −x∈ds), x6=0.

As particles evolve independently, we think of q_x :=q(x,M₀) = P

i≥1ν_x_i(Sxi) as the rate of first fragmentation from the configurationx; the next configurationyhaving lawq(x,dy)/qx. Of course it really makes sense only ifq_x <∞, but this “hold-jump” description can be made rigorous (using a truncation argument) with a small assumption [21, Chapter 1]: that for every ε >0, there exists a constant c_ε >0such that

ν_x(Sx)< c_ε and Z

s((ε,∞))ν_x(ds)< c_εν_x(Sx), x > ε.

It is then checked that X is a pure-fragmentation process. In particular, if ν_x, x > 0, is the image measure of x^αν by the map s 7→ x·s, where α ∈ R and ν is any finite measure on S with R

s((ε,∞))ν(ds) < ∞ for every ε > 0, then the above condition holds, and so does the self-similarity property of Definition 0.1.1; we say that X is the (α, ν)-self-similar fragmentation chain (or simply the ν-homogeneous fragmentation chain when α= 0).

0.1.2.2 Self-similar pure fragmentations

Naturally, aself-similar pure-fragmentation process is a self-similar branching process that is also a pure fragmentation. One example is the (α, ν)-self-similar fragmentation chain of the previous section, where the measure ν was such that ν(S)<∞ and R

s((ε,∞))ν(ds)<∞ for every ε >0. In fact, we can also make sense of a self-similar fragmentation process with

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0.1. FRAGMENTATION PROCESSES 7 infinite dislocation rates. Suppose instead that ν has support in P, that ν({1}) = 0 and

Z

P

(1−p1)ν(dp)<∞. (0.2)

While allowing ν(P) = ∞, condition (0.2) limits the intensity of “macroscopic” dislocations (those given by the mass-partitions which are far from 1). More precisely, if a measurable set A⊂ M₀ is at distance d(1, A)>0 from1, then ν(A)≤d(1, A)⁻¹ R

(1−p₁)ν(dp)<∞.

A fundamental result [22, 17], is that the law P := P₁ of a self-similar P-valued fragmentation process is uniquely determined by its self-similarity index α ∈ R, its erosion coefficient c ∈ [0,∞), and its so called dislocation measure ν, which is a measure on P satisfying to ν({1}) = 0 and (0.2); we shall hence refer to the (α,c, ν)-self-similar fragmentation process (or rather the (c, ν)-homogeneous fragmentation process whenα = 0).

Considering a Poisson point processMonP×Nwith intensityν⊗#, where#is the counting measure onN, the(0, ν)-homogeneous fragmentation processXcan be constructed fromMin such a way thatXonly jumps when some atom(p, k)ofMoccurs, and if it happens at timet, then X(t) is obtained from X(t−) by replacing its k^th largest particle X_k(t−) by the cloud of particles X_k(t−)·p, leaving the other particles unchanged. Erosion simply corresponds to a continuous decay in the fragment masses, to the extent that (exp(−ct)X(t) : t≥0)is a version of the (c, ν)-homogeneous fragmentation process. There also exists some procedure to change the index of self-similarity, transforming the (c, ν)-homogeneous fragmentation process into the (α,c, ν)-self-similar fragmentation. Explaining this transformation as well as the bijection between P and (α,c, ν) requires that we enrich the fragmentation with a genealogical structure. One way to achieve this is via so called interval representations:

one can couple any self-similar pure fragmentation X with some process G onto the usual topology of the open interval (0,1) such that for all s ≤ t, G(t) ⊆ G(s) and X(t) coincides with the non-increasing rearrangements of the lengths of the interval components of G(t).

We refer to [22, Section 3.2] for greater detail.

Example 0.1.4. Let α ∈ R and ν be the law of (U,1−U,0, . . .), where U is uniformly sampled on [¹₂,1). Then the (α,0, ν)-self-similar pure-fragmentation process matches the dividing sandpile process with uniform binary splitting that started this introduction.

0.1.3 Growth-fragmentation processes

Allowing particles to vary continuously between fragmentation events raises intricate questions as well as important applications. The equilibrium between growth and fragmentation has first been studied “deterministically” by analysts. The first stochastic models were introduced recently by Bertoin [18, 19].

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0.1.3.1 Cell systems

We start by constructing a rather general system of non-interacting particles, called cell system in [19]. Our approach is a bit more general and bears similarities with [29, Section 4.2].

Informally, a cell system consists in a family X := {(Xu(t − bu) : bu ≤ t) : u ∈ U} of processes on (0,∞), recorded since their birth times b_u (which we implicitly encode in the notation X_u) and whose negative jumps produce birth events. These processes are assumed to be càdlàg and either converging to or absorbed at 0, so that their negative jumps can be easily enumerated. In [19], thej-th negative jump (ordered by decreasing absolute size) of X_u, occurring say at time t ≥ 0 and with size −y < 0, is the cause of a single daughter cell X_uj born at b_uj := t with X_uj(0) := y. There, processes were indexed by the usual Harris–Ulam tree U :=S

n≥0Nⁿ, where N⁰ := {∅} is reduced to the root of U, which labels the Eve cell X∅.

We can slightly generalize the preceding idea and imagine that the cell material y > 0 lost by X_u at timetduring itsj-th negative jump serves to the creation ofseveral (zero, one, or more) daughter cells X_u,(j,1), X_u,(j,2), etc., all with birth times b_u,(j,k) := t and respective sizes at birthys1,ys2,etc., for some random configurations∈S whose law may depend on both X_u(t−)and y. Thus, we rather choose the indexing set as

U := [

n≥0

(N²)ⁿ = [

n≥0

N²ⁿ.

In this slightly unconventional Harris–Ulam tree, each node u := (u₁, . . . , u_2n) ∈ U at a given generation |u| := n has children ujk := (u₁, . . . , u_2n, j, k), j, k ≥ 1, labelling at the next generation all daughter cells Xujk := Xu,(j,k) originating from the j^th negative jump of X_u, respectively, where these jumps are enumerated in the non-increasing order of their absolute sizes, and chronologically in case ofex aequo. More precisely, if(t₁,−y₁),(t₂,−y₂), . . . denotes this enumeration of negative jump times and sizes, then eithery_j > y_j+1 ory_j =y_j+1 and tj < tj+1, for all j. The children labelled ujk, k ≥ 1, corresponding to the jump (t_j,−y_j), depend on a certain configurationsûj prescribing the relative sizes at birth, namely (X_ujk(0) : k ≥1) =y_j·sûj (when sûj =0, no particle is born). We setb∅ := 0,b_ujk:= b_u+t_j, and agree with the conventions b_ujk := ∞ and X_ujk ··≡ 0 if pûj_k = 0 or X_u has fewer than j negative jumps. Figure 0.3sets up the notation.

The law P_x of X started from a single particle X_∅ with initial size x ≥0 is constructed to ensure the genealogical branching property: for all u ∈ U and i ≥ 1, conditionally on σ((X_u,sûj) :|u| ≤ i, j ≥ 1), the cell systems Xûjk := {X_ujkv: v ∈ U}, |u| = i, j, k ≥ 1, are independent; further, each Xûjk has law P_ysûj

k , where y is the j^th negative jump of Xu

(according to the above enumeration).

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0.1. FRAGMENTATION PROCESSES 9

b∅

X∅

ys²₁

−y

b_2,1

X_2,1

b_1,1

X_1,1

b_1,1,1,1

X_1,1,1,1

Figure 0.3. Illustrating example of cell system.

If Xu has its j^th negative jump −y < 0 at time t, then independent daughter-cells Xujk, k ≥ 1, with respective sizes at birthys^uj_k , k≥1,are born atbujk :=t, wheres^uj is a%(Xu(t−), y/Xu(t−),·)-distributed configuration. (For the picture we chose %≡ δ1, so each cell Xu begets at its j^th negative jump a unique childXuj1, with size at birth equal to the jump size.)

Formally, P_x depends on a probability kernel %(x, r,ds) from (0,∞) × (0,1] to S, the fragmentation kernel, and on a càdlàg Markov process(Y,(P_y)y≥0)on(0,∞), thecell process, which, under the initial distribution P_y, y ≥ 0 (P_y(Y(0) = y) = 1 and P₀(Y ≡ 0) = 1), is either eventually absorbed at 0 or converging to 0 as t → ∞. It is defined with the help of Ionescu-Tulcea’s theorem as the unique distribution on [0,∞)^U such that X_∅ has lawP_xand, conditionally onX∅, the enumeration(t₁,−y₁),(t₂,−y₂), . . .of its negative jumps, and on an independent family (s^j: j ≥ 1) of independent configurations with respective laws %(X∅(tj−), yj/X∅(tj−),·), the cell systems {Xjkv: v ∈ U}, j, k ≥ 1, are independent P_y

js^j_k-distributed variables. We forward the reader to [70] for more rigorous details on the construction of this type of branching processes.

Note that we may as for pure fragmentations give a sense to a dislocation measure ν:

if negative jumps −y < 0 of Y from a certain size x occur at rate θ(x,dy), then particles with sizexfragmentate at rateθ(x,dy)%(x, y/x,ds)into a cloud of particles with initial sizes

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x−y, ys₁, ys₂, ys₃. . .; see the connection in Example 0.1.6. We also underline the “binary case” x 7→ {x−y, y} (for % ≡ δ1), which reduces the construction presented here to that treated in [19].

We callX thecell system driven by (Y, %), but we are in fact more interested in the point process of alive cells:

X(t) := X

u∈U

1^{bu≤t}δXu(t−bu), t≥0.

When (X(t) : t ≥0) is a branching process (in the sense of Definition 0.1.1), it is called the growth-fragmentation process driven by (Y, %). Note that in general, X might very well not be M₀-valued, nor might it enjoy the temporal branching property. However, the existence of some excessive function is sufficient to enforce this. In this direction, write ∆⁻y(s) :=

max(y(s−)−y(s),0) for s > 0, y: [0,∞) → R and suppose that there exist C > 0 and f ∈ B_0< such that for all x >0and t≥0,

E_x

"

C^−tf Y(t)

+ X

0<r≤t

Z

S

X

i≥1

C^−rf ∆⁻Y(r)s_i

%

Y(r−),∆⁻Y(r) Y(r−),ds

#

≤ f(x). (0.3) Then⁴ X is a growth-fragmentation process in Mf. Moreover, f is a C-excessive function for X, and so C^−thX(t), fi, t ≥0, is a supermartingale.

Of course, if X is a growth-fragmentation driven by (Y, %) where the cell process Y has non-increasing sample paths, thenX corresponds to a pure fragmentation. When further the fragmentation kernel % has support in P, we can check that (0.3) trivially holds for C = 1 and f =f₁.

0.1.3.2 Self-similar growth-fragmentations

In this section, we assume that the fragmentation kernel%(x, r,ds)does not depend onx, and by a slight abuse of notation we set%(x, r,ds)≡··%(r,ds). Aself-similar growth-fragmentation process is a growth-fragmentation fulfilling the self-similarity property of Definition 0.1.1. It must be driven by a cell process Y which is itself self-similar (with index α):

for all x >0, the law of (xY(x^αt) : t≥0)under P1 is Px.

Indeed, by [19, Lemma 1], self-similarity then extends to the cell system X driven by (Y, %):

for y > 0 arbitrary and b⁰_u := y^−αb_u, u ∈ U, X_u⁰(t) := yX_u(y^αt), u ∈ U, t ≥ 0, the law of ((X_u⁰, b⁰_u) : u∈U) under P₁ is the same as the law of ((X_u, b_u) :u∈U)under P_y.

4This criterion is easily adapted from [19, Proposition 2 & Theorem 1] proved forC= 1and %≡δ1. By the Markov property, it is fulfilled if and only if the process f ∆⁻Y(r)si

% Y(r−),^∆_Y⁻_(r−)^Y^(r),ds

, t≥0, is a supermartingale underP_x, for everyx >0. WhenY is a semimartingale, this can be investigated for smooth enough functionsf by stochastic calculus.

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0.1. FRAGMENTATION PROCESSES 11 Lamperti [78] characterized all positive self-similar Markov processes (for short, pssMp).

Suppose Y is a pssMp with index α ∈R which is either absorbed at or converging to 0 a.s.

Then θ(t) := inf{s≥0 : Rs

0 Y(r)^αdr > t}, t ≥0, is increasing for t <R∞

0 Y(s)^αds=: ζ, and equals ∞ if t ≥ ζ. Lamperti’s transformation says that under P₁, ξ(t) := logY(θ(t)), t ≥ 0 (with the convention ξ(t) := −∞ for t ≥ ζ), is a Lévy process (a càdlàg process with independent and stationary increments), which is either absorbed at or diverging to −∞.

Conversely, for such a Lévy process ξ, we can define Y as having under P_y the law of (yexp(ξ(τ_y^α_t)) : t ≥ 0), where τ_t := inf{s ≥ 0 : Rs

0 exp(−αξ(r)) dr > t}. In other words, a self-similar cell processY is determined by(α, ξ), whereα ∈Randξis a Lévy process which is either absorbed at or diverging to−∞. In turn,ξis characterized by a quadruple(k, b, σ²,Λ), where k ≥ 0 is the killing rate, b ∈R the drift coefficient, σ² ≥0 the Gaussian component, and Λ is the so calledLévy measure, that is a measure on R\ {0}with R

(1∧y²) Λ(dy)<∞.

Since it describes the intensity of jumps in the cell process, Λ will often be supported on the negative half-line. However, allowing cells to encounter sudden growth can be relevant especially in applications to random planar maps (see [24] or the forthcoming Section 0.3), and it is actually only assumed that R

1^{y>1}e^yΛ(dy) < ∞. Then the distribution of ξ is identified by the Lévy–Khintchine formula E[exp(qξ(t))] = exp(tΨ(q)), t ≥ 0, through a Laplace exponent Ψ of the form

Ψ(q) :=−k+bq+1

2σ²q²+ Z

R

e^qy −1 +q(1−e^y)

Λ(dy), q∈C, (0.4) which makes sense at least for q∈[0,∞). In particular, Ψ : [0,∞)→(−∞,∞] is convex.

In view of Lamperti’s transformation, we shall in this setting refer to Y and X as, respectively, the (α,Ψ)-self-similar cell process and the (α,Ψ, %)-self-similar cell system.

When further X is a branching process (typically, when (0.3) applies), X is called the (α,Ψ, %)-self-similar growth-fragmentation (or rather, if α = 0, the (Ψ, %)-homogeneous growth-fragmentation).

Remark 0.1.5. Different (Ψ, %) may lead to the same self-similar growth-fragmentation process with index α [99].

Example 0.1.6. Letc≥0 and% have support inP. SupposeΛ has support in[−log 2,0) with R

(1∧ |x|) Λ(dx)<∞, and that −ξ is a subordinator, i.e., Ψhas the form Ψ(q) :=−k−cq+

Z

R

(e^qx−1) Λ(dx), q≥0 (0.5)

(which means that Y is non-increasing and never jumps lower than half its current size).

Then the (α,Ψ, %)-self-similar growth-fragmentation is a version of the (α,c, ν)-self-similar

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pure fragmentation, where the dislocation measure ν is given by Z

f(p)ν(dp) = kf(0) + Z Z

f e^x, p1(1−e^x), p2(1−e^x), . . .

Λ(dx)% 1−e^x,dp

, (0.6) for every measurable function f: P → [0,∞). Conversely, suppose ν is a dislocation measure, that is ν({1}) = 0 and R

P(1−p₁)ν(dp)<∞. Then (0.6) holds fork =ν({0}), Λ the image measure of ν by the map p7→ logp₁, and, by disintegration⁵ [72, Corollary 1.23], some probability kernel %(r,dp), r ∈ (0,1], on P which we can interpret as the image measure of ν(· | p₁ = 1−r) by the map p 7→ (_1−p^p^j

1)_j≥2. Further, the (α,c, ν)-self-similar pure fragmentation coincides with the (α,Ψ, %)-self-similar growth-fragmentation with Ψ as in (0.5).

This example shows that the greater generality of self-similar growth-fragmentations with respect to self-similar pure fragmentations lies essentially in the fact that ξneed no longer be the negative of a subordinator; it may for instance have a Brownian component and jumps of unbounded variation.

WhenX is driven by a general(α,Ψ)-self-similar cell processY with Laplace exponentΨ as in (0.4), there is the following specialization of (0.3) to prove excessiveness. Suppose first α = 0, so that Y = xe^ξ under P_x. By stochastic calculus (adapting the proof of [19, Lemma 2]) we have, for all x >0 and t, q ≥0,

E_x

"

Y(t)^q+ X

0<r≤t

Z

S

X

i≥1

(∆⁻Y(r)s_i)^q%

∆⁻Y(r) Y(r−),ds

#

= x^q+x^qκ(q) Z t

0

e^rΨ(q)dr, where

κ(q) := Ψ(q) + Z

(−∞,0)

(1−e^y)^qΛ(dy) Z

S

X

i≥1

s^q_i %(1−e^y,ds), q ≥0. (0.7) Hence (by a supermartingale argument) the inequality (0.3) withκ(q)≤0,f =fqandC = 1 will hold for all x >0 and t ≥ 0, and (by Doob’s optional stopping theorem) even if α 6= 0.

We can here already see that κ, known as the cumulant function, plays a crucial rôle.

Lemma 0.1.7. Suppose q ≥ 0 is such that κ(q) ≤ 0. Then (0.3) holds for f = fq and C = 1, and the process hX(t), f_qi, t ≥0, is a supermartingale under P_x for every x >0.

As a matter of fact, Bertoin and Stephenson [33] have shown that for α 6= 0, the condition of Lemma 0.1.7 is necessary for X to be M0-valued.

5We viewν as a measure on[−∞,0)×P via the bijective transformation p7→(logp1,_1−p^p²

1,_1−p^p³

1, . . .).

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0.1. FRAGMENTATION PROCESSES 13 Naturally, the presence of power functions in additive (super)martingales is expected because of the multiplicative structure of self-similar fragmentations. Let us derive here f(t) :=f(t;x, q) :=E_x[hX(t), f_qi] in the homogeneous case α= 0. Then, the cell process Y under P_x is simply represented as xe^ξ, where ξ is a Lévy process with Laplace exponent Ψ.

By the Lévy–Khintchine formula, the infinitesimal generatorLof Y fulfills Lf_q(x) =x^qΨ(q).

Now, consider the variation of f from f(0) = x^q after an infinitesimal amount of time dt.

On the one hand, the variation due to the growth of the mother particle is Lf_q(x)dt (in the first order). On the other hand, the amount of negative jumps with relative size e^y that have occurred on this time interval is roughly Λ|(−∞,0)(dy) dt, and each begets a random cloud x(1−e^y)·p of daughter particles where p has law %(1−e^y,·), bringing thus a contribution of R P

i≥1(x(1 −e^y)p_i)^q%(1−e^y,dp) to f(dt)−f(0). Putting pieces together, it follows that ∂_tf(0) =x^qκ(q) (at least when κ(q) <∞). Since the branching property easily entails f(t+s) = f(t)f(s)for all s, t ≥0, we conclude that f(t;x, q) = e^tκ(q)x^q. This means that f_q is e^κ(q)-invariant for X, and therefore (by Proposition 0.1.2) e^−tκ(q)hX(t), fqi, t ≥ 0, is a martingale under P_x for every x > 0. We refer to [18, Theorem 1] for a rigorous derivation of this fact.

Example 0.1.8. Let ν be a dislocation measure. With the notation of Example 0.1.6, the cumulant function for the (0, ν)-homogeneous pure-fragmentation process X is

κ(q) = Z

R

(e^qy −1) Λ(dy) + Z

(−∞,0)

(1−e^y)^qΛ(dy) Z

P

X

i≥1

p^q_i %(1−e^y,dp)

= Z Z

[0,∞)×P

x^q−1 + (1−x)^qX

i≥2

p_i 1−p₁

q!

ν(p1 ∈dx)ν(dp|p1 =x)

= Z

P

X

i≥1

p^q_i −1

!

ν(dp).

In particular ifν is conservative, thenκ(1) = 0(and the martingalehX, f1iis trivial). For the example starting this introduction, we haveν(p₁ ∈dx) = 21[¹₂,1)(x) dxandν(p₁+p₂ = 1) = 1, so κ(q) = ^1−q_1+q.

The strictly self-similar case α6= 0 is somewhat different. There, it is generally assumed that Cramér’s hypothesis is satisfied [24, Section 3]:

There exist 0< ω−< ω+ such that κ(ω−) =κ(ω+) = 0 and κ⁰(ω−)>−∞. (0.8) Under this hypothesis, hX, f_ω₊i is a martingale whenα≤0 [24, Corollary 3.5], andhX, f_ω₋i is a uniformly integrable martingale when α≥0[24, Theorem 3.7].

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Remark 0.1.9. Additive martingales cannot be uniformly integrable when α < 0, because the self-similar fragmentation gets eventually extinct almost surely [60, 22, 19]: X(t) = 0 after a P-almost surely finite time t.

Remark 0.1.10(Compensated fragmentation processes; general branching Lévy processes).

• The first construction of homogeneous growth-fragmentations (in [18]) was not made from the point of view of cell systems, but rather directly using projective limits of general branching Lévy processes (see the second point below). More conceptually, one may interpret them as limits of dilated homogeneous pure fragmentations. Suppose indeed that X_n, n ∈ N, are (c_n, ν_n)-homogeneous pure-fragmentation processes such that there is the weak convergence of finite measures on P

(1−p₁)²ν_n(dp) =⇒σ²δ₁(dp) + (1−p₁)²ν(dp), for some σ² ≥0and measure ν onP fulfilling ν({1}) = 0 and

Z

P

(1−p₁)²ν(dp)<∞, (0.9)

and suppose further thatcn−^σ₂² converges asn→ ∞to some constantb ∈R(ifcn ≡0, this is automatically verified with b =−^σ₂²). Then there exist a sequence (d_n)n≥0 of nonnegative numbers and a non-trivial processXsuch that for everyq >2, the convergence in distribution

(exp(d_nt)X_n(t) : t≥0)−−−→^(d)

n→∞ X,

holds in the space of M_f_q-valued càdlàg functions endowed with Skorokhod’s J₁-topology.

Moreover, the dilation coefficients may be chosen as d_n:=R

P(1−p₁)ν_n(dp); if so, thenX is the compensated fragmentation process with characteristics (σ², b, ν). If we disintegrate ν as in (0.6) and defineΨby (0.4) (withk:=ν({0})), then the compensated fragmentation process with characteristics (σ², b, ν)coincides with the (Ψ, %)-homogeneous growth-fragmentation.

• Condition (0.9) is weaker than the integrability requirement (0.2) for the dislocation measure of a self-similar pure fragmentation. When the latter fails, the too strong accumulation of “microscopic” dislocations would instantaneously shatter the mass into dust, so it must be compensated by a suitable dilation of the fragments. This is of course reminiscent of the construction of Lévy processes as compensated Poisson integrals. The analogy is not coincidental [29]: in the same way that Lévy processes characterize infinitely divisible distributions, processes of the form

X

i≥1

δ_log_X_i_(t), t ≥0,

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0.2. ASYMPTOTICS OF SELF-SIMILAR FRAGMENTATIONS 15 whereXis a homogeneous growth-fragmentation, are calledbranching Lévy processes because they identify the random point measures which can be written for any n ≥ 1 as the n-th generation of some branching random walk. These processes are studied in greater detail in [29, 28].

0.2 Asymptotics of self-similar fragmentations

Here, we summarize the large-time asymptotics of several quantities related to homogeneous, self-similar, pure- and growth-fragmentation processes, such as empirical distributions associated with the fragments and the size of the largest fragment.

0.2.1 Homogeneous fragmentations

The bottom line is that for homogeneous fragmentations, the empirical measure of the fragments exhibits a log-Normal distribution.

Theorem 0.2.1 (Kolmogorov [74]). Let X be the ν-homogeneous fragmentation chain, with ν a probability measure onS∩ M_f₀. Suppose thatQ(1) >1andR1

0|logt|³dQ(t)<∞, where for t∈(0,1], Q(t) :=R

#{i∈N: 0< s_i ≤t}ν(ds). Then as t→ ∞, the quantity sup

x∈R

1 hX(t), f₀i

X

i≥1

1^{logXi(t)≤x}−F

x−µt σ√

t

converges to 0in probability, whereF is the cumulative distribution function of the standard Normal distribution, µ:= Q(1)⁻¹R1

0 logtdQ(t), andσ² := Q(1)⁻¹R1

0(logt−µ)²dQ(t).

Nowadays, Kolmogorov’s theorem should be seen as a version of a central limit theorem for branching random walks [9, 10, 40]. Applying additive martingale techniques due to Biggins [39, 41], Bertoin [23] and Bertoin and Rouault [32] then established asymptotics for possibly infinite dislocation measures:

Theorem 0.2.2 (Bertoin [23], [21, Theorem 1.2]). Letνhave support inP withν({1}) = 0 and R

(1−p₁)ν(dp)<∞, and let X be the(0, ν)-homogeneous pure-fragmentation process.

Suppose that the cumulant function κ(q) :=

Z

S

X

i≥1

s^q_i −1

!

ν(ds), q≥0, (0.10)

has a (necessarily unique) zero ω ≥0 and that, for somer >1, Z

S

X

i≥1

s^ω_i

!r

ν(ds)<∞. (0.11)

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Then as t→ ∞, the empirical measures X

i≥1

X_i(t)^ωδ¹

tlogXi(t) and X

i≥1

X_i(t)^ωδ^√

t ¹_tlogXi(t)−µ (0.12) converge in L¹(P) to M∞δµ and M∞·σN respectively, in the sense of weak convergence of measures, whereµ:= κ⁰(ω),σ² :=κ⁰⁰(ω),N is a standard Gaussian random variable, andM∞

is the terminal value of the uniformly integrable martingale hX, f_ωi.

The local central limit theorem for branching random walks (Stone [101], Biggins [41]) specializes to homogeneous pure fragmentations as follows.

Theorem 0.2.3 (Bertoin and Rouault [32]). Let the dislocation measure ν, the homogeneous pure-fragmentation X, and the cumulant function κ as above, and suppose further thatν is conservative and non-geometric. Then for every Riemann integrable function f: (0,∞)→R with compact support, there is as t→ ∞ the P-almost sure convergence of

q 7−→√

te⁻ ^κ(q)−qκ⁰^(q) tX

i≥1

f

X_i(t)e^−κ⁰^(q)t toward

q 7−→ M∞(q) p2πκ⁰⁰(q)

Z ∞ 0

f(y) y^q+1 dy,

locally uniformly in U := {q ≥ 0 : |κ(q)| < ∞ and qκ⁰(q)−κ(q) < 0}, where M_∞(q) is the terminal value of the uniformly integrable martingale e^−tκ(q)hX(t), f_qi, t ≥0.

Asymptotics for the largest fragmentX₁(t) are also tractable:

Proposition 0.2.4 (Bertoin [23]). Assuming further thatU has no empty interior, we have

t→∞lim 1

t logX₁(t) =κ⁰(¯q)

P-almost surely on the non-extinction event {∀t ≥0 :X(t)6=0}, where q¯:= supU.

We show in Chapter 1 that Theorem 0.2.3 and Proposition 0.2.4 also hold when X is a compensated fragmentation process. Actually, we augment Proposition 0.2.4 by deriving further asymptotic orders for the largest fragment X1 using results that have since appeared in the literature on branching random walks [3, 100].

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0.2. ASYMPTOTICS OF SELF-SIMILAR FRAGMENTATIONS 17 Theorem 0.2.5

LetX be the compensated fragmentation with characteristics (σ², b, ν) and cumulant function

κ(q) := 1

2σ²q²+bq+ Z

P

∞

X

i=1

p^q_i −1 +q(1−p₁)

!

ν(dp), q≥0.

Suppose κ(0) > 0, ν non-geometric and ν p₂ > 0, P

i≥1p^q_i > 1

< ∞ whenever κ(q)<∞ and q <1. Then:

• The conclusion of Theorem 0.2.3 holds.

• If U := {q ≥0 : κ(q)<∞ and qκ⁰(q)−κ(q)<0} has no empty interior, then there exist a constant C^∗ > 0 and a nonnegative random variable D∞ such that, for every x >0,

t→∞lim P

t^3/2¯^qe^−κ⁰^(¯^q)tX₁(t)≤x

=E

e^−C^∗^D^∞^/x . Moreover, D∞ >0 P-almost surely on the non-extinction event.

0.2.2 Self-similar fragmentations

Recall that a self-similar fragmentation process with negative index gets almost surely extinct.

In this section, we summarize asymptotics of self-similar fragmentations when α > 0. As before, we are interested in empirical measures of the fragments and in the largest fragment.

Theorem 0.2.6 (Bertoin and Gnedin [26]). LetX be the (α, ν)-self-similar fragmentation chain with index α > 0 and non-geometric dislocation measureν on S with ν(S) = 1and ν({0}) = 0. Suppose that

κ(q) :=

Z

S

X

i≥1

s^q_i −1

!

ν(ds), q∈C, has a positive root ω∈(0,∞), and define the random finite measures

ρ_t:=X

i≥1

X_i(t)^ωδ_t1/αXi(t), t≥0.

(i) If µ := κ⁰(ω) < ∞, then as t → ∞, ρ_t converges in P-mean to a probability measure ρ∞, in the sense of weak convergence of measures. Setting φ(q) := −κ(q), the limit ρ∞ is uniquely determined by the moments

Z ∞ 0

y^kαρ_∞(dy) = (k−1)!

αµ φ(ω+α)· · ·φ(ω+α(k−1)), k ≥1. (0.13)

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(ii) If

¯q := inf{q ∈ R: κ(q) < ∞} < ω and (0.11) holds for r = 2, then for every ε > 0 and every measurable function f: (0,1] → R dominated by f

¯q−ω+ε, there is as t → ∞ the convergence in L²(P) of hρ_t, fi toward M∞hρ∞, fi, where M∞ is the terminal value of the uniformly integrable martingale hX, f_ωi.

Example 0.2.7. Suppose ν(p₁ ∈dx) = 21[¹₂,1)(x) dx and ν(p₁+p₂ = 1) = 1, so κ(q) = ^1−q_1+q with¯q =−1<1 =ω,µ= ¹₂, andρ∞is the law ofG^1/α whereGis Gamma(2/α)-distributed.

Consequently, the average density u_t(dx) := E[hX(t), f₀i]⁻¹E[X(t)(dx)] of particles with sizexat a large timetapproximately follows the generalized Gamma(t^−1/α,1, α)-distribution E[G^−1/α]⁻¹E[G^−1/α; (G/t)^1/α∈dx]. This is consistent with the observations of Filippov [60];

0.3 Self-similar fragmentations as scaling limits

Self-similarity stipulates ascale invariance property in time and space. Self-similar processes thus naturally “attract” limits of rescaled dynamics. Doubtless the most classical example is that of Brownian motion, which as proved by Donsker [53] is the continuous limit of rescaled random walks with finite variance. A few years later, Lamperti [78] fully characterized all real Markov processes that arise as weak limits of suitably normalized processes (so called scaling limits). Among these, and besides Brownian motion, are notably stable processes, stable Lévy processes, Bessel processes, stable Lévy processes conditioned to stay positive, etc. For more recent results on self-similar Markov processes, see the survey [92].